- Email: [email protected]
Constrained systems and Grassmannians
Nuclear Physics B264 (1986) 317-336 © North-Holland Publishing Company
CONSTRAINED SYSTEMS AND GRASSMANNIANS* Laurent BAULIEU
The Rockefeller Unioersit); 1230 York Avenue, New York, N Y 10021, USA and L P T H E Physique Theoretique, Paris Tour 16 4, Place Jussieu F 75230, Paris, France
Bernard GROSSMAN t
The Rockefeller University, 1230 York Avenue, New York, N Y 10021, USA Received 14 March 1984
We develop a formalism for the construction of the phase space of all constrained hamiltonian systems. This formalism is manifestly covariant and shows how one can understand anomalies as obstructions in this phase space.
Dirac [1] has established a beautiful formalism for the quantization of a hamiltonian system for which the phase-space variables are not all independent. Such constrained hamiltonian systems can occur in two different ways. Either there is not a one-to-one correspondence between the (q, p) variables of the hamiltonian formalism and the (q, ~) variables of the lagrangian (as in gauge field theories) or the phase-space variables are subject to external constraints (as in the case of a particle moving in a potential but restricted to a submanifold of the phase space of the position variables.) Dirac's method has been historically important in the consistent quantization of Yang-Mills theories. It was transposed by Faddeev and Popov [2] into the functional formalism. The lack of explicit covariance in Dirac's quantization method is a disadvantage. By definition, it reduces the study of a hamiltonian system with a given phase space (q, p) to that of a modified system for which the phase space is a submanifold which is generally gauge dependent. Furthermore, it leaves aside the notion of anomalies [3]. This notion is deeply rooted in the geometry, as has now been well-established in Yang-Mills theory, one of the essential .examples of a constrained system. In an interesting series of papers, Batalin, Fradkin, Fradkina, and Vilkovisky [4] have noted that Dirac's formalism can be generalized by introducing ghost degrees * In the memory of P.A.M. Dirac. * Work supported in part under the Department of Energy Contract Grant Number DE-AC0281ER40033B. 317
318
L. Baulieu. B. Grossman / Constrained systems
of freedom. Furthermore, a structure analogous to the BRS symmetry [5] emerges to play a crucial role in the lagrangian formalism of gauge symmetries. In this article, we extend these notions and point out that the geometrical nature of the Dirac method along with the possibilities of anomalies is clarified when the phase space of constrained systems is suitably enlarged. We shall be concerned only with systems with first-class constraints since the case of second-class constraints only present inessential technical complications. Consider a hamiltonian Ho(q i, pi), where the canonical variables (q~, Pi), 1 ~< i ~
(2)
(H0,+°)p= V;+'.
(3)
The symbol ( )p means the Poisson bracket in P (X,Y)p=
Op, Oq~ ( -
Oq' Op,]'
(4)
where g(X) is the Lorentz grading of X. (g(X)= 0(1) if X is bosonic (fermionic).) The quantities f~"O,Vff are in general dependent on the variables in P. Eq. (2) means that the constraints q)" form an "algebra" with structure constants, fv"~, which are in general phase-space dependent in order for eq. (1) to be consistent. Eq. (3) is essential in showing that the constraints are stable under time evolution. The geometrical interpretation of V~" is obtained by realizing that the constrained physical space may be curved. This will be made explicit in one of our examples. The constraints q)" reduce the dimension of the physically allowed phase space from 2n to at most 2n - m. However, within the 2n - m dimensional phase space there is an m-parameter family of phase-space trajectories (flows) generated by ~" which are stable under time evolution [6]. In order to specify a representative 2 ( n - m) dimensional physical phase space one must specify m additional conditions, called secondary constraints. Each equation specifies a curve, X" = 0, which must cut each of the flows in one point. The physical dynamics, however, must be independent of the functional form for X". This is implemented in the formalism by promoting to a dynamical variable, )t", the Lagrange multiplier for constraints, d?~. This requires the introduction of ~r~, the momenta conjugate to X~, as well as ghost coordinates (adjoined variables c ", d , (with ghost number 1)) and g,, d ~ (with ghost number - 1). The bars indicate we have complexified the Grassmann algebra. With the introduction of ?t" and ~r", we have in fact enlarged the phase space from 2n to
L. Baulieu, B. Grossman / Constrained systems
319
2(n + m) dimensions. In unitary gauge, ~r~ will be the Lagrange multiplier for the secondary constraints. The introduction of the ghost degrees of freedom, which are anticommuting Grassmann variables with effectively negative degrees of freedom, then reduces the phase space to 2(n - m) dimensions. In more mathematical terms, we would like to consider a fibration of the 2(n + m) dimensional cotangent bundle (determined by (q, p) and (~, ~r)) by the 4m-dimensional complexified cotangent bundle determined by the algebra of constraints (determined by the (c, d) and (g, aT)). In fact this structure has been developed with the hope that we do have a fibration of a bundle. We therefore have two quartets constituting a 2(n + m) dynamical-variable phase space representing the effective 2 ( n - m) dimensional phase space 6~. 62 . q (i) o, ca(l) /
do(_i)/P'(O)~doO) ~H~(O)/
\?.( - i)
~x.(0) / 1~<
1~<
l <~i<~n
l <~a<~m
(5)
( ) determines the ghost (grassmannian) number. We also have the graded Poisson bracket in 6~ defined as
{X,Y}e=
OX OX E ~p~ OQb ( ?,,,Q,,
)g(x)g(x) OX
OY ] -OQ" - - - Opb ,
(6)
~9
yielding
{ pi, p,}~=O= ( qi, qi} e;,
(7a)
{ pi, qJ}~=8/,
(7b)
{X~, ~ } ~ = 0
= { ~r~,~r#} 9,
{ X., ~rt~} ~ = 8~B,
(7c) (7d)
( c°,c ~ ) 9 = 0 = ( do, d~ ) 9,
(re)
{eo,e~}9=0= { d",d~} 9,
(7f)
{ c a, d# } 9 = 8 f = { ~ , d t~} 9"
(7g)
320
L, Baulieu, B. Grossman / Constrained.~ystem~'
The ghost number, g(X), of each phase-space variable, X, is specified in eq. (5). The ghost number of the product of fields is the sum of the ghost numbers. For X, Y phase-space variables we have commutativity
{ x, r
= (
r, x
(8)
We will assume the Jacobi identity and then try to prove consistency [7]. In order to obtain a well-defined hamiltonian system, we must have a ghost number zero function of all phase-space variables which generates a one-parameter family of translations in time t. This function, called the hamiltonian, determines a differential 1-form d f for any given function f of all the phase-space variables X by
of { H, f } =
Y'~
2 0---X
(9a)
X= qi, Pi ,~", II. ca, ~,~,d °, d~
- (X, d f ) ,
(9b)
so that
Of
d f = • - ~ dX
(lOa)
#f = Y'~ -~ k d t OX '
(lOb)
where we have suppressed variables of summation in the latter formulas. As a consequence of the above, we can define an operator D=d-
{H, },
(lOc)
which annihilates all phase-space variables. In addition, we require the existence of a nilpoint BRS symmetry in order to compensate the apparent redundancy in the enlarged phase space of dynamical variables. Clearly, this symmetry must change the ghost number. We shall denote the BRS symmetry as implemented on the algebra of phase space by s={I2,
}~,
(11)
where $2 is an as yet unknown function with ghost number one. Furthermore, s must be the analogue of d, but in this case it is differentiation along a grassmannian direction 8. We thus consider the function 0
s=dO~-~- {[2,
}9
(12)
L. Baulieu, B. Grossman / Constrained systems
321
which annihilates all phase space variables. ~2 is interpreted as generating translations along unphysical directions. We now have that s 2 = O, which is equivalent to the nilpotency condition on 12 {52, 52} ~ = 0.
(13)
This is a nontrivial condition since I2 is a grassmannian 1-form in phase space. We also have that
(H,H}¢=O,
d2=O.
(14)
Finally, we have that sd+ds=0,
(15a)
{ 12, H ) = 0,
(15b)
if
because by the Jacobi identity, we have for any X
{O,{H,X}}e+(H,{I2, X}~,}={{~2, H},X}~.
(16)
Therefore, the hamiltonian is defined only up to the BRS transformation of some composite operator since for any X {[2, H + {~2, X}}
(16a)
= {12, H } + {J2, {~2, X } )
(16b)
= 0 + 1{{12, I2}, X} = 0.
(16c)
The preceding discussion can be summarized by considering the operator b = D + S,
(17)
which must annihilate all phase-space variables. Consistency demands that D 2 = 0 ¢ * {~2,52} = { H , H } - ~5{£2, H} = 0 .
(18)
and H must have ghost number one and zero, respectively; using dimensionality and conservation of ghost number, we find that there is a minimal solution to (18) for which H has a quadratic dependence and ~ a cubic dependence on ghost
322
L. Baulieu, B. Grossman / Constrained svstems
degrees of freedom.
H=H°(Pi'qi)+d"Vflc'+
(
( xo)}
/2'(g~d~)-X~
(19a)
= Ho( Pi, q*) + d.V~ cB + d.d" + )t% + s(8"X.),
(19b)
/2 = %a7 ~ + c ~ - ±r 2 ~ rF~r/~,4 '~ "-~ ~'v"
(20)
/2 has been determined by an expansion in terms of ghost n u m b e r + 1 whose coefficients are determined by the nilpotency condition. This d e t e r m i n e s / 2 up to a t r a n s f o r m a t i o n of rr --+ a~r + fl?~. The demonstration of the satisfaction of eq. (18) we leave to the appendices• Because d 2 = s 2 - ds + sd = 0, we have a double differential complex [8]. Namely, we can define the differential forms, /2*.0 on phase space with a 2(n + m ) dimensional basis for 1-forms defined by dqi, dpi, d?,~,d% as well as /20.* on the g r a s s m a n n i a n space of ghosts and antighosts with a 4m-dimensional complex basis for 1-forms defined by c ~, d ~, g~, d e such that an n-form in /20,* has ghost n u m b e r n. Morever, we can define forms in both phase space and G r a s s m a n n directions/2".* so that /2r, t is an r-form in phase space with ghost n u m b e r t. We would like to consider a flat space with no non-trivial topology (i.e. we would like acyclicity) so that we have a double differential complex with exact sequences, i.e.
1' ...+ ~ 2 ( n + m ) ,
(for r =~ O)
d / 2 r ' t = o ~ r , t = d / 2 r-l't ,
(21a)
(fort=g0)
s/2r't=O~/2r"=s[gr"-l,
(21b)
1' - m ...+
_+/22(n+m),
1' T _..>/21, - 1
$ __.+~'~O,--m
$ ~
•
•
•
...+ ~"~0, --1
T 1 ~
~2(n+m),O
1' T _.+ / 2 1 , 0
d$ S ff~O,O ...+
1' ...+ / 2 2 ( n + m ) , l
1" __+/22(n+m),m_.+
__+
1'
T
T _.+ /21,1
...+
1' S ~?0,1 ---+
$ -.-a.
.
.
.
.__> ~? O, m _.+
.
(22) N o t i c e that the arrows go on indefinitely because we are considering differential
L. Baulieu, B. Grossman / Constrained systems
323
forms that m a y be nonlocal. We can also consider the cohomology groups ker(s:120,i ---, I2°, i+1)
Hi(s) = I m ( s : I2°'i-~ ~ I2°'i) '
(23a),
ker(d: J2i'° ~ $2i+1'°) Hi(d) = I m ( d : I2i-]'°--* ~T'°) "
(23b)
F o r a flat space we would have Hi(d) = 0 for i > 0. If the structure constants f ~ v are those for the Lie algebra of a compact Lie group, then in a gauge field theory H~(s) for 0 ~< i ~< n determines the cohomology of the gauge group (consisting of m a p s of space-time into the Lie group) because ¢,* ~,a.,~, s e a ~ _ !2J~8"y" L. .
(24)
This is the Maurer-Cartan equation for the differential forms on the gauge group, provided s 2 = 0. As familiar examples with non-trivial topology, one could choose a non-trivial two-sphere S 2 with non-trivial H2(d) in space with a background monopole field corresponding to a non-trivial Hi(s) or a nontrivial S 4 in spacetime with a background instanton H3(s):# 0). The important complex to consider is
(25) r=0
with k e r ( d + s: ~2" ---, I2 " + l )
H " ( d + s) = I r a ( d + s: a2" - 1 ~ ~2~) "
(26)
This is because eqs. (18) followed from the consistency condition for the operator
D=D+S. N o w an obstruction to the construction of a flat (trivial) phase space is a non-trivial element of H(k+l)(d + s) for k > 0. This means that [9] (dw
s ) [ A k + l ' O + A k ' l + " ' " + A0'k+l]---0
(27a)
for mi'j E Qi'J(i +j = k + 1) such that [A k ÷ l ' ° + - - . + A °'k+]] ~ I m ( d + s ) .
(27b)
324
L. Baulieu, B. Grossrnan / Constrained systems
S/tk + a.O = _ dA k, 1
(28b)
sAk, 1 = -- d A k - 1,2,
(28c)
s~ l'k = - d ~ °'k+l ,
(28d)
sk °'k+l = 0.
(28e)
This non-trivial element of H k+ l ( d + s ) is an obstruction to the construction of the classical phase space. Given any function of the phase-space variables f ( q , p, X, ~r) which is homotopic to the identity, there is a h o m o t o p y operator T [8, 9] so that Td + d T = 1 - f *
(29a)
where f * is a cochain map induced by f. One can think of f as determining a canonical transformation, so one would hope that there would be no obstruction to this h o m o t o p y when one consideres the constraints on the phase space. Therefore, one would like (d+s)T+
T(d+ s) = 1 -f*.
(29b)
However, zeroes of d + s are obstructions to the existence of such a T. When there are no zeroes of d + s, we have s T + Ts = 0.
(29c)
When there are zeroes, there is a cocycle 0 (dO = 0) which measures the obstruction so that Ak+l'° = TO, dA k+l'° = dTO = ( d T + T d ) O = 0.
(29d) (29e)
In plain English, suppose we consider a loop in (q, p, ~, ~r) parameterized by ~'. If we are to have a globally hamiltonian system, this loop determines a 1-parameter family of flows generated by the constraints. The existence of a nonzero Ak+~ prevents the lifting of the constraints from local generators to global vectors [10]. The theory we get is inconsistent because 8H has a nonvanishing matrix element between states of different ghost numbers: one can create ghosts in the physical Hilbert space! In a non-abelian gauge theory, An'l is the non-abelian anomaly. ~ is the generator of local gauge transformations. The fact that they do not lift to global vector fields is stated in mathematical language by saying An'l determines the first
L. Baulieu, B. Grossman / Constrained systems
325
obstruction cocycle of the gauge group. In fact, as we shall reexamine later, it has been shown that eqs. (28a-c) determine a tower of obstruction cocycles that ends with the non-trivial element A°' ,+1 ~ H,+l(s ) for the Lie group of the theory. There can be an anomaly in n space-time dimensions if there is a generator of S "+1 in the cohomology of the Lie group. In fact we need only consider k = n because the bundle of the big space is contractible, i.e. f* --- 0. We note that such a generator can appear even with trivial background fields unlike the generator of S"-1 in n-dimensions (the instanton). We point out that Gribov (and more generally, Singer [11]) described another ambiguity that arises in gauge fixing, an ambiguity that also depends on the non-trivial topology of the gauge group. The Gribov or gauge fixing ambiguity is the fact that one cannot choose a global section of the bundle ~ ~ ~ ~ / ~ (where ~ is the gauge group and ~ the set of vector potentials) because ~ and ~ / ~ are not topologically trivial even though ~ is contractible. In contrast, an anomaly is an obstruction to lifting the action of ~ as a local gauge transformation to a unitary operator on functionals of ~. As a final remark on the cohomology H*(d + s), we note that by a technique very similar to that for proving {I2, H } = 0, if X ~ $2°'° such that ( d + s ) X = O, (in a theory with constant V), then X can be expressed as X = .~o + ~r~T? - q~.Tz~ ,
(30a)
such that This implies that in an anomaly free theory H*(d + s)= H ° ( d + s) and if there are no obstructions (such as additional conserved quantities) to setting ~r~ = 0 = q~, we can do so. The proof, which we leave for appendix C, depends on a minimal expansion for X, commutativity, and the Jacobi identity. Of course, if there are additional conserved quantities like angular momentum for a spherical potential, these must be fixed before setting % = 0 = q~. If we have an anomaly-free theory, then we can hope to obtain a consistent perturbative expansion for the theory. Formally, one easily obtains unitarity for the anomaly-free theory. Since the dynamics is independent of the form for X (i.e., H is defined up to the BRS transformation of an operator, which is the only place X enters), we can continuously vary X by canonical transformation to a form so that the ghost has no interactions. ?~ and ~r are then explicitly Lagrange multipliers of and X respectively. In this unitary gauge, the ghosts decouple since the hamiltonian depends only on 2(n - m) variables. Therefore, the dynamics is manifestly unitary. Of course the form and the existence of X* in a unitary gauge depends upon q~. One must only demand that {q,, X*} is independent of (qi, pi). We will now consider three examples: the quantum mechanics of a particle confined to a circle on a two sphere, quantum electrodynamics in four dimensions, and a non-abelian gauge theory in four dimensions. Each example will display an anomaly. In the first case, it will be a magnetic monopole. In QED, one can have the
326
L. Baulieu. B. Grossman / Constrainedsystems
abelian a n o m a l y of Adler-Bell and Jackiw. A non-abelian gauge theory can have a local or global anomaly. First consider the hamiltonian for a charged particle on a two-sphere* 1
2
H=½(P~+s--:~nzoP, ) + s ( g d ) ( _ ~ ) , s = cp~,+ d~r,
(31a) (31b)
sH = 0 = s 2 .
(31c)
T h e choice of s forces the particle not to have motion in the ~ direction. Suppose F = gO( r )sinO dO /x dq),
(32)
where we have written O(r) so that the two-sphere can be considered as the b o u n d a r y of a 3-volume with additional coordinate r. The 3-dimensional space c a n n o t be continued to r = 0 and we obtain the obstruction cocycle d F [12] such that
d F = g S ( r ) s i n O d r /x dO/x dq~,
(s + d ) d f = O,
(338)
s d F = 0 ~ s F = dA,
(33c)
A = cgO (r)(constant - cos 0 ) , sA = O,
(33a)
r 4: O.
(33d) (33e)
Because of the structure of the two-sphere we must choose the customary two patches so that A is non-singular, i.e.
AX=cg(1 - c o s 0 ) A II= - c g ( l + c o s O ) A I - A ll = 2go
-
i e
0~0<½7r+e, ½1r + e < 0...< ~r,
at 0 -- ½~r
sU(ep),
U( dp) = e 2iegee.
(34a) (348) (34c) (34d) (34e)
W e see n o w the origin of the anomaly F. We must choose a gauge fixing term X so that [X, P , } 4:0. ff or any smooth non-constant function of d? will do. This is equivalent to choosing a particular value of ft. Since the non-singular connection * Since R3 - {0} is homotopicaUy S2, we might as well consider S2.
L. Baulieu, B. Grossman / Constrained systems
327
associated with the magnetic monopole requires a nontrivial gauge transformation U(¢) at the equator, we cannot consistently fix the gauge and make the gauge transformation. In a quantum field theoretical generalization of this construction, we can see how the magnetic monopole in SU(5) GUT broken down to SU(3) x SU(2) x U(1) is an obstruction leaving an unobstructed SU(2)× U(1) [13]. To define the theory with two patches on S 2 requires a gauge transformation at the equator that is generated by a diagonal matrix ( 0 , 0 , 1 , - 1, 0) with the first three elements of the diagonal corresponding to SU(3), the last two SU(2). The generators of SU(3) × SU(2) gauge transformations do not all commute with this patching gauge transformation. There is therefore the above-mentioned obstruction to fixing a gauge. This example can be turned around in the following way. We can consider the momentum p~ as generating an action (translation) on the equatorial circle. One might like to know whether this action can be extended to the entire two-sphere. We have found that there exists a possible obstruction to this extension determined by *r3(S2) = Z, i.e. one can twist the S1 or U(1) action of p, with S 2 to obtain fibration of S 3. This is the famous Hopf fibration. We mention that there exists a similar hamiltonian description of the Hopf fibrations of the spheres S 7 -'~ S 4 and S15 --, S 8 as well as of CPN, S 2N+1 ~ CP N [14]. Before going on to the next example, it is instructive to show a related problem gives an example of a non-vanishing Vfl Ho = ½( p~ + ~ p ~
+ ~ p1 ,
2) ,
(35a) (35b)
s = cr + dTr,
1 (36c)
Silo = cpr = r c - P r , r
1
V = -Pr.
(35d)
r
We see that V is a consequence of the curvilinear coordinates. For QED
[(ea)( ~1
2
-x
(36a)
+s2)+ x xo,e,+ad+e(o,e,,x}c,
(36c)
S = qrd-- 8 i E i c , sm i~-- - S i c , s E i = O,
AI = c E . B ,
s c = 0, sc = ~ ,
(36b)
sd = - O i E i , s d = O,
sTr = O,
SJk = d ,
(36d) (36e) (37a)
328
L. Baulieu, B. Grossman /
Constrained systems
so that ( i-i, z~~ } = ( e , x e . n }
(37b)
= dE. B .
(37c)
A priori, A1 could depend upon d; however, there is no solution to the anomaly equation proportional to d. In this case, the well-known fact that A 0 is not a dynamical variable requires the constraint on the generator of local gauge transformations, q~, = o,E, = o.
(38)
The Lagrange multiplier ~ = A 0 is promoted to a dynamical variable with momentum conjugate ~r. If we choose the axial gauge X =A3, { O i E i ( x ) , A3(y)} = - a 3 8 ( x - y ) so this gauge is ghost-free and the theory is unitary provided that there is no anomaly. It is also well-known that there is an inconsistency in QED, as a result of the anomaly, if one tries to gauge a representation of fermions that is not vector-like. This has a nice interpretation if we imagine the U(1) gauge theory imbedded in a spontaneously broken SU(2) gauge theory with one chiral SU(2) fermion doublet. This theory has magnetic monopoles whose ground state would acquire an electric charge in the presence of C P violation if there were no massless fermions. Since U(1) is unbroken, the chiral fermions are massless and the chiral symmetry allows one to rotate the electric charge of the magnetic monopole to zero. This is a flat direction in the gauge orbit space. Finally, we have for QCD H = ~Tr(E ~+ ~2) + s(~.dO)
-X °
,
(39a)
= ½Tr( E 2 + B 2) + ~r"x, ~ - 2%( DIE[' + f~'Bdacr)
+ dad ° + eo( D,E;, x o )
(39b)
s = qr, d-" - DiEi~c ° - ~ f 2 f c ~ d ,
(39c)
sAT=
-Oi ca ,
S~a=d
sEi~ = - f f t ~ c a E i ~ , SCa = q'ga ,
a ,
s~r. = O,
sda=
-OiEia-ff~dflc
(40b)
yL, ff,.. ,
A1 = A 4,1 = n o n - a b e l i a n ( H , a 4'1 } ---=s AS'0 .
(40a)
s d " = O,
l_ ¢ a B ,4 ,~v
s e a = -- 2 J
~,
anomaly,
(41a) (41b)
L. Baulieu, B. Grossman / Constrained systems
329
The description for QCD is very similar to that for QED. Notice that the acyclicity in ~, ~r, c, d corresponds to our choice for a flat space and non-trivial principal bundle. (For an instanton field, the abelian anomaly matters, however.) Once again the anomaly is non-vanishing when certain representations of fermions are gauged. As a result, the physical space is not gauge-invariant, i.e. it is inconsistent with the imposition of the constraint of Gauss' law q~ = D,E,,~ + j o = 0
(42)
(where j o is the non-abelian fermonic charge). Instead, the action of a gauge transformation g:
A ----,Ag= gAg-X + gdg -1 ,
(43)
when implemented by a unitary operator U(g) on a functional of A such as the determinent of the Dirac operator det ~0 becomes U( g ) ~( A ) = e"~(A; g)qJ(A g) with ~1
"~-
(44)
JA~ a 1-cocycle where the integration is over both the group space and
space-time and U ( g l ) U ( g 2 ) ~ ( A) = ei~'2(A;g~,g2)U(glg2)6( A ) ,
(45)
defined so that sA] = - d A 2. The existence of a 2-cocycle a 2 [15] has recently been shown to be equivalent to the existence of an anomalous commutator for the generators of local gauge transformations. In four dimensions, the descent equations (28) imply the existence of 1-cocycle to 5-cocycles. This means that there is a topological cohomology theory with elements a, ~ H"(G) [16] (or a similar theory for H " ( G ) with G the Lie algebra) so that a, maps n elements of the group into a complex phase C*, a.:
(gl . . . . . g . ) ~ a . ( A ; g l
. . . . . g.)
(46)
and the coboundary operator is 8: H" ---, H n+l for
3a.(gl ..... g.) = ~ ( - l i ) a . ( g l ..... gi... g.),
(47)
i=0
One can easily show that if H* = HI(G,C*), then one can describe the universal covering group of the group G as a direct sum of G and C*; i.e. G and the phases defined by the first cocycle. If there is also a non-trivial second cocycle, then the universal covering space is a twisted product of G and C*, twisted by the existence
330
L. Baulieu, B. Grossman / Constrained systems
of a 2. In both cases, one can describe the relation between G, (3" and the universal covering space G by the exact sequence i
1 ~ C* ~
~
G ~
G ~ 1,
(48)
where in the case of the existence of only a 1-cocycle, i is invertible and G = C* • G. If the m a x i m u m non-trivial cohomology is H 2 ( G , C * ) , then one can lift to the central extension by adding only one additional generator to the algebra. The redefined commutation relations satisfy the algebra of constraints (2). We can see this very easily as follows on the Lie algebra levels. Define
a( G A, G") = [G A, G"] --fA"cGC ,
(49)
This is a 2-cocycle. It defines a vector field X,~(A ' B) [10] such that (50a) = [XxA, Xx.]
-fC.xx,
(50b)
=0,
since the vector field associated with the generator G A is the same locally as the vector field in the tangent space of the group in the direction M. Moreover
E
a([[M,)~"],hc]) =ra=O.
(51)
cyclic permutations
Therefore a is a constant locally and also a 2-cocycle. One defines the central extension by adding a generator of the real numbers r ~ R with the Lie bracket
[v A + r, a" + s] = [v A, G"] + . ( a A, a")
(52)
and the projection 7r: G A + r ~ G A. The equivalence classes of central extensions are in one-to-one correspondence with the second cohomology group H2(G, C*). The relevance of this argument to our description of anomalies is that if H 3 ( G , C *) is non-vanishing, then there is an obstruction to the construction of a central extension with a finite number of new generators. In conclusion, we have extended the consideration of constrained hamiltonian systems to include grassmannian variables associated with the algebra of constraints. Anomalies in gauge theories have been understood as a type of obstruction cocycle in the non-trivial cohomology of d + s. This cohomology also includes quantum mechanical invariance of the motion and characteristic classes associated with non-trivial fibrations as in the magnetic monopole.
L. Baulieu, B. Grossman / Constrainedsystems
331
M o s t of this w o r k was c o m p l e t e d while one of us (L.B.) was visiting Rockefeller U n i v e r s i t y in the fall of 1984. H e thanks Professor N. K h u r i for the h o s p i t a l i t y a n d s u p p o r t o f the Rockefeller theoretical physics group. W e w o u l d also lie to t h a n k P r o f e s s o r J a m e s Stasheff for his interest and n u m e r o u s discussions.
Note added in proof A f t e r this w o r k was c o m p l e t e d we b e c a m e aware of two related p a p e r s b y H e n n e a u x [18] a n d M c M u l l a n [19].
Appendix A NILPOTENCY OF S
- : { ,~. /:~ } :c,e. + ¼::a,{ ::~. : j ' }
c°':'a,.
+ f,a~'c'~c#f v,~P'c"'d#, .
(A.1)
F r o m (2) we have that
(cO,o. c% } =zo;::,. { eOa, ( eo,~,coo} } c% "ca=O
(A.2)
by Jacobi identity
(A.3)
= { ~., L~'*, } :::
= { ~,,. L ; } q,pc°c ~
(A.4)
+L;c"c"I. :c % ( COs, f,¢~' ) c%'~c~ -f,#°c"caf,,'t°c '~'= O.
(a.5:
Finally,
c'~c#d.t( f~a, f~'~, } c"'c#'d~,, =
0
(A.6)
* While eq. (A.5) is true for the cases considered in the text, it is not true in the generic case, e.g. supergravity and a relativistic membrane [17]. We thank Dr. M. Henneaux for pointing this out to us.
332
L. Baulieu, B. Grossman / Constrained systems
by symmetry. From (2)
(2)
{ ¢,~,¢/~ } = f¢/i~v,
(:~,0.(:.:)) = (:%. :-,~)+~ +f'Pv{ f'oo, fir },
(a.7)
-((:.:u}:) eo~},eop } byJacobi.
+{{fj,
(a.8)
Multiply by grassmannian c, d:
c~cOdo{fa.°,f<#i.:} d:~'c"cp
:c.c.,°-:o,.( iu.+.) + ((+'.i.:):) + { { fn/', q¢~} q>tl } l c<~ctl
=::a.[-lo,,{ ij.~>'}
(A.9)
+
{ iU::..~> °)
-
{ iUio',, ~,/'}] <°<"
(A.IO)
= cacOd. [f.arf~o'f,", - fn°'fn/'f<,.~,
-fnp'f#"~'f,~.~, + fa/'f,~°~'fl~..i, + f,~"'~f,/'flt..~,] c<'clt =0.
(A.11) Appendix B
BRS I N V A R I A N C E OF H
We define the following quantities: qA=( X q~~ ) , x a = ( X ~ )- x o
~rA= (p:r~),
,
a° =
(B.O
(B.2)
(,~o¢o),
(B.3) 14= 14o + % v ; ¢ -
{ %x a , o : ~ - ~ <1: :r r b
~
c d
n ),
(B.4)
333
L Baulieu, B. Grossman / Constrained systems
with V"b and U~be determined by V~B and f~ar with the appropriate components equal to zero. ~2 = Gb~ b - ~1 U h~d p b)),))~,
(B.5)
{~2, H ) ='O'~{G,,, Ho} + ~(Ho, Ub,.a} Pfll"~l a + G~V"b~) b 1 lib
-
UbcdO~b~CVde'lje
c dr~ I/'a
+ ~ "~ cd~l rl ~ ~-- b
- c2, { ~2, V#b ) .qb.
(8.6)
We use (B.7)
{ G,,, H o ) = - V~bGb,
(8.8)
( ~, H } = ~ U%~oV°?~,7 ~- ~ U , } % V / ¢ , f -- Ubcd°~b]]CVde~ e + Ubceg~@b~C~] d
+ { o,, v~ )e~n'n~- °L{ ~, vo~} ¢
(B.9)
__I -- ~6~a{ UCde , Vab } 6~c~dne,b.
We have the following identity
( H o, {G#, Gb} ) -- ( n o, U , b " ) G , + U,,b"{Ho,G,} =(Gb,(G~,Ho}}+(G,,,(Ho,Gb} = - ( G b, V~CG~) + (O,,, Vb~G,.} =
_
(B.10) )
(B.11) (B.12)
UhJG~Vo~ + U~fV~G~
- { G b, V J ) G ~ +
{G,,, Vb~}O~
= UabCVcdGd q- ( Ho, U,6 c } Go,
(B.13) (8.14)
{ Ho, Cobc } = - U b f V J + U o j V / - uo#v;
- { Gb, V~~ } + { G o, V~c }.
(B.15)
(B.15) serves as an integrability condition resulting from the Jacobi identity, i.e.
334
L. Baulieu, B. Grossman / Constrainedsystems
insist that ( [2, H o } = - ( J2, P . V ~ b ) such that { H o, G. } = Vh.Gh. Then the above identity must hold. By symmetry the following must be zero.
( Vo/, V; } - ( vo;, vj } + (u~/, v;} - { <,:, v/ ) --~ ( Ubj, Vae } -- ( Ubde, V/ } -- ( Ub/, Vde } "q- ( Ubae, V j ) --{UaJ, Vbe}--[-(Udae V/}--(Ud/,Vae}'~-(Ubde,
V/}=O.
(U.16)
Similarly for ( f-~v, f-'a'Y') = 0.
Appendix
C
GAUGE INVARIANCE OF fjo,o Suppose X = X°( p, q) + p 17"~(p, q) 71b ,
(c.1)
H = H°(p,
(c.2)
q) + 6).V"b~b,
V = constant. Supressing indices, we have: {H, X} = 0 =, (a) (b)
( H °, X ° ) = 0 ,
(C.3a)
~2(H°,~}~ + ~[V,~]~ =0
(C.3b) (C.4)
i.e. (b) ~ ( H °, ~" } = - V~',
(G, (u°, P}} = -(G, Vf/} = ( < (/~°,G}) + {H°, (C,~}) = - (G, v ) ~ -
V(G,f~)
(c.5) (c.6)
= (~,v~) + (H°,(6,~)}.
(c.7)
(H°,(G,I/))
(C.8)
Then =0,
335
L. Baulieu, B. Grossman / Constrained svstems
implies ( G, ~" ) = const = 0,
(~, x )
(C.9)
= {6, x ° ) n + a v n + ( v , x °) ~
+ ~(a,
+ WPrl~'r/+ 6)lT:wrlrl,
P)~ (C.10) (C.11a)
( ao, x ° } = - a f ' , (U, X ° ) = V U - V~'+ 17"U+ (G, I7:) - (G, 1,7"),
(6o, x } = (Go, x ° ) = - G f <
(C.11b)
(c.a2)
Eq. (C.11b) is just an integrability condition for (C.11a) and proved by the Jacobi identity. But this is equivalent to X = T ° + T"G,,
(C.13)
with
(G~,X°)=(6~,T°6°)=-6~.
(C.14)
By the above, choose V = V. This choice for V means (3) satisfies an integrability so Z given, V, we can solve for T ". QED. References [1] P.A.M. Dirac, Lectures on quantum mechanics (Yeshiva University, New York 1964) [2] L. Faddeev and V.N. Popov, Phys. Lett. B25 (1967) 30 [3] D. Gross and R. Jackiw, Phys. Rev. D6 (1972) 477; W. Bardeen, Phys. Rev. 184 (1969) 1843; P.H. Frampton and T.W. Kephart, Phys. Rev. Lett. 50 (1983) 1343, 1347; L. Baulieu, Nucl. Phys. B241 (1983) 55; B. Zumino, Y.S. Wu and A. Zee, Nucl. Phys. B239 (1969) 1459; E. Witten, Phys. Lett. l17B (1982) 324; L. Alvarez-Gaum~ and E. Witten, Nucl. Phys. B234 (1983) 269; L. Alvarez-Gaum6 and P. Ginsparg, Nucl. Phys. B243 (1984) 499; M. Atiyah and I. Singer, Proc. Nat. Acad. Sciences, USA 81 (1984) 2599; O. Alvarez, I.M. Singer and B. Zumino, Commun. Math. Phys. 96 (1984) 409 [4] E.S. Fradkin G.A. Vilkovisky, Phys. Lett. 55B (1975) 229; E.S. Fradkin T.E. Fradkina, Phys. Lett. 72B (1978) 343; I.A. Batalin E.S. Fradkin, Phys. Lett. 86B (1979) 263 [5] C. Becchi, A. Rouet R. Stora, Phys. Lett. 52B (1974) 344 [6] C. Itzykson J. Zuber, Quantum field theory (McGraw-Hill, 1980) [7] D. Sullivan, Differential computations in topology, Institut des Hautes Etudes Scientifique [8] R. Bott and L. Tu, Differential forms in algebraic topology (Springer, New York, 1982) [9] R. Bott, Differential topology, foliations and Gelfand-Fuks Cohomology (Springer, Berlin 1978); Relatively, groups and topology II (Les Houches 1983) ed. by B.S. De Witt and R. Stora (North Holland, 1984): Lectures given by L. Baulieu and B. Zumino and R. Stora
336
L. Baulieu, B. Grossman / Constrained systems
[10] P. Simms and N. Woodhouse, Lectures on geometric quantization (Springer, Berlin, 1976) [11] V.N. Gribov, SLAC translation 176 (1977); I. Singer, Commun. Math. Phys. 60 (1963) 7 [12] B. Grossman, Phys. Lett. B152 (1985) 93; R. Jackiw, Phys. Rev. Lett. 54 (1985) 159; Y.S. Wu and A. Zee, Phys. Lett. B152 (1985) 98; D. Boulware, S. Deser and B. Zumino, Phys. Lett. B, in press. [13] A. Aboulsaood, Phys. Lett. 125B (1983) 467; Nucl. Phys. B226 (1983) 309; P. Nelson and A. Manohar, Phys. Rev, Lett. 50 (1983) 943; A.P. Balachandran, et. al., Phys. Rev. Lett. 50 (1983) 1553 [14] B. Grossman, J. Stasheff and T.W. Kephart, Commun. Math. Phys. 96 (1984) 431 [15] L. Faddeev, Phys. Lett. 145B (1984) 81; B. Zumino, ITP-UCSB-preprint NSF-ITP-84-150; J. Mickelson, University of Helsinki preprint, HU-TFT-83-57 Commun. Math. Phys., to be published [16] K. Brown, Cohomology of groups (Springer, New York 1982); J. Stasheff, Bulletin AMS 84 (1978) 513 [17] M. Henneaux, Phys. Rev. Lett. 55 (1985) 769 [18] M. Henneaux, Univ. of Texas preprint, to be published in Phys. Reports [19] D. McMullan, Imperial College preprint 83-84/21 (1984)
CONSTRAINED SYSTEMS AND GRASSMANNIANS* Laurent BAULIEU
The Rockefeller Unioersit); 1230 York Avenue, New York, N Y 10021, USA and L P T H E Physique Theoretique, Paris Tour 16 4, Place Jussieu F 75230, Paris, France
Bernard GROSSMAN t
The Rockefeller University, 1230 York Avenue, New York, N Y 10021, USA Received 14 March 1984
We develop a formalism for the construction of the phase space of all constrained hamiltonian systems. This formalism is manifestly covariant and shows how one can understand anomalies as obstructions in this phase space.
Dirac [1] has established a beautiful formalism for the quantization of a hamiltonian system for which the phase-space variables are not all independent. Such constrained hamiltonian systems can occur in two different ways. Either there is not a one-to-one correspondence between the (q, p) variables of the hamiltonian formalism and the (q, ~) variables of the lagrangian (as in gauge field theories) or the phase-space variables are subject to external constraints (as in the case of a particle moving in a potential but restricted to a submanifold of the phase space of the position variables.) Dirac's method has been historically important in the consistent quantization of Yang-Mills theories. It was transposed by Faddeev and Popov [2] into the functional formalism. The lack of explicit covariance in Dirac's quantization method is a disadvantage. By definition, it reduces the study of a hamiltonian system with a given phase space (q, p) to that of a modified system for which the phase space is a submanifold which is generally gauge dependent. Furthermore, it leaves aside the notion of anomalies [3]. This notion is deeply rooted in the geometry, as has now been well-established in Yang-Mills theory, one of the essential .examples of a constrained system. In an interesting series of papers, Batalin, Fradkin, Fradkina, and Vilkovisky [4] have noted that Dirac's formalism can be generalized by introducing ghost degrees * In the memory of P.A.M. Dirac. * Work supported in part under the Department of Energy Contract Grant Number DE-AC0281ER40033B. 317
318
L. Baulieu. B. Grossman / Constrained systems
of freedom. Furthermore, a structure analogous to the BRS symmetry [5] emerges to play a crucial role in the lagrangian formalism of gauge symmetries. In this article, we extend these notions and point out that the geometrical nature of the Dirac method along with the possibilities of anomalies is clarified when the phase space of constrained systems is suitably enlarged. We shall be concerned only with systems with first-class constraints since the case of second-class constraints only present inessential technical complications. Consider a hamiltonian Ho(q i, pi), where the canonical variables (q~, Pi), 1 ~< i ~
(2)
(H0,+°)p= V;+'.
(3)
The symbol ( )p means the Poisson bracket in P (X,Y)p=
Op, Oq~ ( -
Oq' Op,]'
(4)
where g(X) is the Lorentz grading of X. (g(X)= 0(1) if X is bosonic (fermionic).) The quantities f~"O,Vff are in general dependent on the variables in P. Eq. (2) means that the constraints q)" form an "algebra" with structure constants, fv"~, which are in general phase-space dependent in order for eq. (1) to be consistent. Eq. (3) is essential in showing that the constraints are stable under time evolution. The geometrical interpretation of V~" is obtained by realizing that the constrained physical space may be curved. This will be made explicit in one of our examples. The constraints q)" reduce the dimension of the physically allowed phase space from 2n to at most 2n - m. However, within the 2n - m dimensional phase space there is an m-parameter family of phase-space trajectories (flows) generated by ~" which are stable under time evolution [6]. In order to specify a representative 2 ( n - m) dimensional physical phase space one must specify m additional conditions, called secondary constraints. Each equation specifies a curve, X" = 0, which must cut each of the flows in one point. The physical dynamics, however, must be independent of the functional form for X". This is implemented in the formalism by promoting to a dynamical variable, )t", the Lagrange multiplier for constraints, d?~. This requires the introduction of ~r~, the momenta conjugate to X~, as well as ghost coordinates (adjoined variables c ", d , (with ghost number 1)) and g,, d ~ (with ghost number - 1). The bars indicate we have complexified the Grassmann algebra. With the introduction of ?t" and ~r", we have in fact enlarged the phase space from 2n to
L. Baulieu, B. Grossman / Constrained systems
319
2(n + m) dimensions. In unitary gauge, ~r~ will be the Lagrange multiplier for the secondary constraints. The introduction of the ghost degrees of freedom, which are anticommuting Grassmann variables with effectively negative degrees of freedom, then reduces the phase space to 2(n - m) dimensions. In more mathematical terms, we would like to consider a fibration of the 2(n + m) dimensional cotangent bundle (determined by (q, p) and (~, ~r)) by the 4m-dimensional complexified cotangent bundle determined by the algebra of constraints (determined by the (c, d) and (g, aT)). In fact this structure has been developed with the hope that we do have a fibration of a bundle. We therefore have two quartets constituting a 2(n + m) dynamical-variable phase space representing the effective 2 ( n - m) dimensional phase space 6~. 62 . q (i) o, ca(l) /
do(_i)/P'(O)~doO) ~H~(O)/
\?.( - i)
~x.(0) / 1~<
1~<
l <~i<~n
l <~a<~m
(5)
( ) determines the ghost (grassmannian) number. We also have the graded Poisson bracket in 6~ defined as
{X,Y}e=
OX OX E ~p~ OQb ( ?,,,Q,,
)g(x)g(x) OX
OY ] -OQ" - - - Opb ,
(6)
~9
yielding
{ pi, p,}~=O= ( qi, qi} e;,
(7a)
{ pi, qJ}~=8/,
(7b)
{X~, ~ } ~ = 0
= { ~r~,~r#} 9,
{ X., ~rt~} ~ = 8~B,
(7c) (7d)
( c°,c ~ ) 9 = 0 = ( do, d~ ) 9,
(re)
{eo,e~}9=0= { d",d~} 9,
(7f)
{ c a, d# } 9 = 8 f = { ~ , d t~} 9"
(7g)
320
L, Baulieu, B. Grossman / Constrained.~ystem~'
The ghost number, g(X), of each phase-space variable, X, is specified in eq. (5). The ghost number of the product of fields is the sum of the ghost numbers. For X, Y phase-space variables we have commutativity
{ x, r
= (
r, x
(8)
We will assume the Jacobi identity and then try to prove consistency [7]. In order to obtain a well-defined hamiltonian system, we must have a ghost number zero function of all phase-space variables which generates a one-parameter family of translations in time t. This function, called the hamiltonian, determines a differential 1-form d f for any given function f of all the phase-space variables X by
of { H, f } =
Y'~
2 0---X
(9a)
X= qi, Pi ,~", II. ca, ~,~,d °, d~
- (X, d f ) ,
(9b)
so that
Of
d f = • - ~ dX
(lOa)
#f = Y'~ -~ k d t OX '
(lOb)
where we have suppressed variables of summation in the latter formulas. As a consequence of the above, we can define an operator D=d-
{H, },
(lOc)
which annihilates all phase-space variables. In addition, we require the existence of a nilpoint BRS symmetry in order to compensate the apparent redundancy in the enlarged phase space of dynamical variables. Clearly, this symmetry must change the ghost number. We shall denote the BRS symmetry as implemented on the algebra of phase space by s={I2,
}~,
(11)
where $2 is an as yet unknown function with ghost number one. Furthermore, s must be the analogue of d, but in this case it is differentiation along a grassmannian direction 8. We thus consider the function 0
s=dO~-~- {[2,
}9
(12)
L. Baulieu, B. Grossman / Constrained systems
321
which annihilates all phase space variables. ~2 is interpreted as generating translations along unphysical directions. We now have that s 2 = O, which is equivalent to the nilpotency condition on 12 {52, 52} ~ = 0.
(13)
This is a nontrivial condition since I2 is a grassmannian 1-form in phase space. We also have that
(H,H}¢=O,
d2=O.
(14)
Finally, we have that sd+ds=0,
(15a)
{ 12, H ) = 0,
(15b)
if
because by the Jacobi identity, we have for any X
{O,{H,X}}e+(H,{I2, X}~,}={{~2, H},X}~.
(16)
Therefore, the hamiltonian is defined only up to the BRS transformation of some composite operator since for any X {[2, H + {~2, X}}
(16a)
= {12, H } + {J2, {~2, X } )
(16b)
= 0 + 1{{12, I2}, X} = 0.
(16c)
The preceding discussion can be summarized by considering the operator b = D + S,
(17)
which must annihilate all phase-space variables. Consistency demands that D 2 = 0 ¢ * {~2,52} = { H , H } - ~5{£2, H} = 0 .
(18)
and H must have ghost number one and zero, respectively; using dimensionality and conservation of ghost number, we find that there is a minimal solution to (18) for which H has a quadratic dependence and ~ a cubic dependence on ghost
322
L. Baulieu, B. Grossman / Constrained svstems
degrees of freedom.
H=H°(Pi'qi)+d"Vflc'+
(
( xo)}
/2'(g~d~)-X~
(19a)
= Ho( Pi, q*) + d.V~ cB + d.d" + )t% + s(8"X.),
(19b)
/2 = %a7 ~ + c ~ - ±r 2 ~ rF~r/~,4 '~ "-~ ~'v"
(20)
/2 has been determined by an expansion in terms of ghost n u m b e r + 1 whose coefficients are determined by the nilpotency condition. This d e t e r m i n e s / 2 up to a t r a n s f o r m a t i o n of rr --+ a~r + fl?~. The demonstration of the satisfaction of eq. (18) we leave to the appendices• Because d 2 = s 2 - ds + sd = 0, we have a double differential complex [8]. Namely, we can define the differential forms, /2*.0 on phase space with a 2(n + m ) dimensional basis for 1-forms defined by dqi, dpi, d?,~,d% as well as /20.* on the g r a s s m a n n i a n space of ghosts and antighosts with a 4m-dimensional complex basis for 1-forms defined by c ~, d ~, g~, d e such that an n-form in /20,* has ghost n u m b e r n. Morever, we can define forms in both phase space and G r a s s m a n n directions/2".* so that /2r, t is an r-form in phase space with ghost n u m b e r t. We would like to consider a flat space with no non-trivial topology (i.e. we would like acyclicity) so that we have a double differential complex with exact sequences, i.e.
1' ...+ ~ 2 ( n + m ) ,
(for r =~ O)
d / 2 r ' t = o ~ r , t = d / 2 r-l't ,
(21a)
(fort=g0)
s/2r't=O~/2r"=s[gr"-l,
(21b)
1' - m ...+
_+/22(n+m),
1' T _..>/21, - 1
$ __.+~'~O,--m
$ ~
•
•
•
...+ ~"~0, --1
T 1 ~
~2(n+m),O
1' T _.+ / 2 1 , 0
d$ S ff~O,O ...+
1' ...+ / 2 2 ( n + m ) , l
1" __+/22(n+m),m_.+
__+
1'
T
T _.+ /21,1
...+
1' S ~?0,1 ---+
$ -.-a.
.
.
.
.__> ~? O, m _.+
.
(22) N o t i c e that the arrows go on indefinitely because we are considering differential
L. Baulieu, B. Grossman / Constrained systems
323
forms that m a y be nonlocal. We can also consider the cohomology groups ker(s:120,i ---, I2°, i+1)
Hi(s) = I m ( s : I2°'i-~ ~ I2°'i) '
(23a),
ker(d: J2i'° ~ $2i+1'°) Hi(d) = I m ( d : I2i-]'°--* ~T'°) "
(23b)
F o r a flat space we would have Hi(d) = 0 for i > 0. If the structure constants f ~ v are those for the Lie algebra of a compact Lie group, then in a gauge field theory H~(s) for 0 ~< i ~< n determines the cohomology of the gauge group (consisting of m a p s of space-time into the Lie group) because ¢,* ~,a.,~, s e a ~ _ !2J~8"y" L. .
(24)
This is the Maurer-Cartan equation for the differential forms on the gauge group, provided s 2 = 0. As familiar examples with non-trivial topology, one could choose a non-trivial two-sphere S 2 with non-trivial H2(d) in space with a background monopole field corresponding to a non-trivial Hi(s) or a nontrivial S 4 in spacetime with a background instanton H3(s):# 0). The important complex to consider is
(25) r=0
with k e r ( d + s: ~2" ---, I2 " + l )
H " ( d + s) = I r a ( d + s: a2" - 1 ~ ~2~) "
(26)
This is because eqs. (18) followed from the consistency condition for the operator
D=D+S. N o w an obstruction to the construction of a flat (trivial) phase space is a non-trivial element of H(k+l)(d + s) for k > 0. This means that [9] (dw
s ) [ A k + l ' O + A k ' l + " ' " + A0'k+l]---0
(27a)
for mi'j E Qi'J(i +j = k + 1) such that [A k ÷ l ' ° + - - . + A °'k+]] ~ I m ( d + s ) .
(27b)
324
L. Baulieu, B. Grossrnan / Constrained systems
S/tk + a.O = _ dA k, 1
(28b)
sAk, 1 = -- d A k - 1,2,
(28c)
s~ l'k = - d ~ °'k+l ,
(28d)
sk °'k+l = 0.
(28e)
This non-trivial element of H k+ l ( d + s ) is an obstruction to the construction of the classical phase space. Given any function of the phase-space variables f ( q , p, X, ~r) which is homotopic to the identity, there is a h o m o t o p y operator T [8, 9] so that Td + d T = 1 - f *
(29a)
where f * is a cochain map induced by f. One can think of f as determining a canonical transformation, so one would hope that there would be no obstruction to this h o m o t o p y when one consideres the constraints on the phase space. Therefore, one would like (d+s)T+
T(d+ s) = 1 -f*.
(29b)
However, zeroes of d + s are obstructions to the existence of such a T. When there are no zeroes of d + s, we have s T + Ts = 0.
(29c)
When there are zeroes, there is a cocycle 0 (dO = 0) which measures the obstruction so that Ak+l'° = TO, dA k+l'° = dTO = ( d T + T d ) O = 0.
(29d) (29e)
In plain English, suppose we consider a loop in (q, p, ~, ~r) parameterized by ~'. If we are to have a globally hamiltonian system, this loop determines a 1-parameter family of flows generated by the constraints. The existence of a nonzero Ak+~ prevents the lifting of the constraints from local generators to global vectors [10]. The theory we get is inconsistent because 8H has a nonvanishing matrix element between states of different ghost numbers: one can create ghosts in the physical Hilbert space! In a non-abelian gauge theory, An'l is the non-abelian anomaly. ~ is the generator of local gauge transformations. The fact that they do not lift to global vector fields is stated in mathematical language by saying An'l determines the first
L. Baulieu, B. Grossman / Constrained systems
325
obstruction cocycle of the gauge group. In fact, as we shall reexamine later, it has been shown that eqs. (28a-c) determine a tower of obstruction cocycles that ends with the non-trivial element A°' ,+1 ~ H,+l(s ) for the Lie group of the theory. There can be an anomaly in n space-time dimensions if there is a generator of S "+1 in the cohomology of the Lie group. In fact we need only consider k = n because the bundle of the big space is contractible, i.e. f* --- 0. We note that such a generator can appear even with trivial background fields unlike the generator of S"-1 in n-dimensions (the instanton). We point out that Gribov (and more generally, Singer [11]) described another ambiguity that arises in gauge fixing, an ambiguity that also depends on the non-trivial topology of the gauge group. The Gribov or gauge fixing ambiguity is the fact that one cannot choose a global section of the bundle ~ ~ ~ ~ / ~ (where ~ is the gauge group and ~ the set of vector potentials) because ~ and ~ / ~ are not topologically trivial even though ~ is contractible. In contrast, an anomaly is an obstruction to lifting the action of ~ as a local gauge transformation to a unitary operator on functionals of ~. As a final remark on the cohomology H*(d + s), we note that by a technique very similar to that for proving {I2, H } = 0, if X ~ $2°'° such that ( d + s ) X = O, (in a theory with constant V), then X can be expressed as X = .~o + ~r~T? - q~.Tz~ ,
(30a)
such that This implies that in an anomaly free theory H*(d + s)= H ° ( d + s) and if there are no obstructions (such as additional conserved quantities) to setting ~r~ = 0 = q~, we can do so. The proof, which we leave for appendix C, depends on a minimal expansion for X, commutativity, and the Jacobi identity. Of course, if there are additional conserved quantities like angular momentum for a spherical potential, these must be fixed before setting % = 0 = q~. If we have an anomaly-free theory, then we can hope to obtain a consistent perturbative expansion for the theory. Formally, one easily obtains unitarity for the anomaly-free theory. Since the dynamics is independent of the form for X (i.e., H is defined up to the BRS transformation of an operator, which is the only place X enters), we can continuously vary X by canonical transformation to a form so that the ghost has no interactions. ?~ and ~r are then explicitly Lagrange multipliers of and X respectively. In this unitary gauge, the ghosts decouple since the hamiltonian depends only on 2(n - m) variables. Therefore, the dynamics is manifestly unitary. Of course the form and the existence of X* in a unitary gauge depends upon q~. One must only demand that {q,, X*} is independent of (qi, pi). We will now consider three examples: the quantum mechanics of a particle confined to a circle on a two sphere, quantum electrodynamics in four dimensions, and a non-abelian gauge theory in four dimensions. Each example will display an anomaly. In the first case, it will be a magnetic monopole. In QED, one can have the
326
L. Baulieu. B. Grossman / Constrainedsystems
abelian a n o m a l y of Adler-Bell and Jackiw. A non-abelian gauge theory can have a local or global anomaly. First consider the hamiltonian for a charged particle on a two-sphere* 1
2
H=½(P~+s--:~nzoP, ) + s ( g d ) ( _ ~ ) , s = cp~,+ d~r,
(31a) (31b)
sH = 0 = s 2 .
(31c)
T h e choice of s forces the particle not to have motion in the ~ direction. Suppose F = gO( r )sinO dO /x dq),
(32)
where we have written O(r) so that the two-sphere can be considered as the b o u n d a r y of a 3-volume with additional coordinate r. The 3-dimensional space c a n n o t be continued to r = 0 and we obtain the obstruction cocycle d F [12] such that
d F = g S ( r ) s i n O d r /x dO/x dq~,
(s + d ) d f = O,
(338)
s d F = 0 ~ s F = dA,
(33c)
A = cgO (r)(constant - cos 0 ) , sA = O,
(33a)
r 4: O.
(33d) (33e)
Because of the structure of the two-sphere we must choose the customary two patches so that A is non-singular, i.e.
AX=cg(1 - c o s 0 ) A II= - c g ( l + c o s O ) A I - A ll = 2go
-
i e
0~0<½7r+e, ½1r + e < 0...< ~r,
at 0 -- ½~r
sU(ep),
U( dp) = e 2iegee.
(34a) (348) (34c) (34d) (34e)
W e see n o w the origin of the anomaly F. We must choose a gauge fixing term X so that [X, P , } 4:0. ff or any smooth non-constant function of d? will do. This is equivalent to choosing a particular value of ft. Since the non-singular connection * Since R3 - {0} is homotopicaUy S2, we might as well consider S2.
L. Baulieu, B. Grossman / Constrained systems
327
associated with the magnetic monopole requires a nontrivial gauge transformation U(¢) at the equator, we cannot consistently fix the gauge and make the gauge transformation. In a quantum field theoretical generalization of this construction, we can see how the magnetic monopole in SU(5) GUT broken down to SU(3) x SU(2) x U(1) is an obstruction leaving an unobstructed SU(2)× U(1) [13]. To define the theory with two patches on S 2 requires a gauge transformation at the equator that is generated by a diagonal matrix ( 0 , 0 , 1 , - 1, 0) with the first three elements of the diagonal corresponding to SU(3), the last two SU(2). The generators of SU(3) × SU(2) gauge transformations do not all commute with this patching gauge transformation. There is therefore the above-mentioned obstruction to fixing a gauge. This example can be turned around in the following way. We can consider the momentum p~ as generating an action (translation) on the equatorial circle. One might like to know whether this action can be extended to the entire two-sphere. We have found that there exists a possible obstruction to this extension determined by *r3(S2) = Z, i.e. one can twist the S1 or U(1) action of p, with S 2 to obtain fibration of S 3. This is the famous Hopf fibration. We mention that there exists a similar hamiltonian description of the Hopf fibrations of the spheres S 7 -'~ S 4 and S15 --, S 8 as well as of CPN, S 2N+1 ~ CP N [14]. Before going on to the next example, it is instructive to show a related problem gives an example of a non-vanishing Vfl Ho = ½( p~ + ~ p ~
+ ~ p1 ,
2) ,
(35a) (35b)
s = cr + dTr,
1 (36c)
Silo = cpr = r c - P r , r
1
V = -Pr.
(35d)
r
We see that V is a consequence of the curvilinear coordinates. For QED
[(ea)( ~1
2
-x
(36a)
+s2)+ x xo,e,+ad+e(o,e,,x}c,
(36c)
S = qrd-- 8 i E i c , sm i~-- - S i c , s E i = O,
AI = c E . B ,
s c = 0, sc = ~ ,
(36b)
sd = - O i E i , s d = O,
sTr = O,
SJk = d ,
(36d) (36e) (37a)
328
L. Baulieu, B. Grossman /
Constrained systems
so that ( i-i, z~~ } = ( e , x e . n }
(37b)
= dE. B .
(37c)
A priori, A1 could depend upon d; however, there is no solution to the anomaly equation proportional to d. In this case, the well-known fact that A 0 is not a dynamical variable requires the constraint on the generator of local gauge transformations, q~, = o,E, = o.
(38)
The Lagrange multiplier ~ = A 0 is promoted to a dynamical variable with momentum conjugate ~r. If we choose the axial gauge X =A3, { O i E i ( x ) , A3(y)} = - a 3 8 ( x - y ) so this gauge is ghost-free and the theory is unitary provided that there is no anomaly. It is also well-known that there is an inconsistency in QED, as a result of the anomaly, if one tries to gauge a representation of fermions that is not vector-like. This has a nice interpretation if we imagine the U(1) gauge theory imbedded in a spontaneously broken SU(2) gauge theory with one chiral SU(2) fermion doublet. This theory has magnetic monopoles whose ground state would acquire an electric charge in the presence of C P violation if there were no massless fermions. Since U(1) is unbroken, the chiral fermions are massless and the chiral symmetry allows one to rotate the electric charge of the magnetic monopole to zero. This is a flat direction in the gauge orbit space. Finally, we have for QCD H = ~Tr(E ~+ ~2) + s(~.dO)
-X °
,
(39a)
= ½Tr( E 2 + B 2) + ~r"x, ~ - 2%( DIE[' + f~'Bdacr)
+ dad ° + eo( D,E;, x o )
(39b)
s = qr, d-" - DiEi~c ° - ~ f 2 f c ~ d ,
(39c)
sAT=
-Oi ca ,
S~a=d
sEi~ = - f f t ~ c a E i ~ , SCa = q'ga ,
a ,
s~r. = O,
sda=
-OiEia-ff~dflc
(40b)
yL, ff,.. ,
A1 = A 4,1 = n o n - a b e l i a n ( H , a 4'1 } ---=s AS'0 .
(40a)
s d " = O,
l_ ¢ a B ,4 ,~v
s e a = -- 2 J
~,
anomaly,
(41a) (41b)
L. Baulieu, B. Grossman / Constrained systems
329
The description for QCD is very similar to that for QED. Notice that the acyclicity in ~, ~r, c, d corresponds to our choice for a flat space and non-trivial principal bundle. (For an instanton field, the abelian anomaly matters, however.) Once again the anomaly is non-vanishing when certain representations of fermions are gauged. As a result, the physical space is not gauge-invariant, i.e. it is inconsistent with the imposition of the constraint of Gauss' law q~ = D,E,,~ + j o = 0
(42)
(where j o is the non-abelian fermonic charge). Instead, the action of a gauge transformation g:
A ----,Ag= gAg-X + gdg -1 ,
(43)
when implemented by a unitary operator U(g) on a functional of A such as the determinent of the Dirac operator det ~0 becomes U( g ) ~( A ) = e"~(A; g)qJ(A g) with ~1
"~-
(44)
JA~ a 1-cocycle where the integration is over both the group space and
space-time and U ( g l ) U ( g 2 ) ~ ( A) = ei~'2(A;g~,g2)U(glg2)6( A ) ,
(45)
defined so that sA] = - d A 2. The existence of a 2-cocycle a 2 [15] has recently been shown to be equivalent to the existence of an anomalous commutator for the generators of local gauge transformations. In four dimensions, the descent equations (28) imply the existence of 1-cocycle to 5-cocycles. This means that there is a topological cohomology theory with elements a, ~ H"(G) [16] (or a similar theory for H " ( G ) with G the Lie algebra) so that a, maps n elements of the group into a complex phase C*, a.:
(gl . . . . . g . ) ~ a . ( A ; g l
. . . . . g.)
(46)
and the coboundary operator is 8: H" ---, H n+l for
3a.(gl ..... g.) = ~ ( - l i ) a . ( g l ..... gi... g.),
(47)
i=0
One can easily show that if H* = HI(G,C*), then one can describe the universal covering group of the group G as a direct sum of G and C*; i.e. G and the phases defined by the first cocycle. If there is also a non-trivial second cocycle, then the universal covering space is a twisted product of G and C*, twisted by the existence
330
L. Baulieu, B. Grossman / Constrained systems
of a 2. In both cases, one can describe the relation between G, (3" and the universal covering space G by the exact sequence i
1 ~ C* ~
~
G ~
G ~ 1,
(48)
where in the case of the existence of only a 1-cocycle, i is invertible and G = C* • G. If the m a x i m u m non-trivial cohomology is H 2 ( G , C * ) , then one can lift to the central extension by adding only one additional generator to the algebra. The redefined commutation relations satisfy the algebra of constraints (2). We can see this very easily as follows on the Lie algebra levels. Define
a( G A, G") = [G A, G"] --fA"cGC ,
(49)
This is a 2-cocycle. It defines a vector field X,~(A ' B) [10] such that (50a) = [XxA, Xx.]
-fC.xx,
(50b)
=0,
since the vector field associated with the generator G A is the same locally as the vector field in the tangent space of the group in the direction M. Moreover
E
a([[M,)~"],hc]) =ra=O.
(51)
cyclic permutations
Therefore a is a constant locally and also a 2-cocycle. One defines the central extension by adding a generator of the real numbers r ~ R with the Lie bracket
[v A + r, a" + s] = [v A, G"] + . ( a A, a")
(52)
and the projection 7r: G A + r ~ G A. The equivalence classes of central extensions are in one-to-one correspondence with the second cohomology group H2(G, C*). The relevance of this argument to our description of anomalies is that if H 3 ( G , C *) is non-vanishing, then there is an obstruction to the construction of a central extension with a finite number of new generators. In conclusion, we have extended the consideration of constrained hamiltonian systems to include grassmannian variables associated with the algebra of constraints. Anomalies in gauge theories have been understood as a type of obstruction cocycle in the non-trivial cohomology of d + s. This cohomology also includes quantum mechanical invariance of the motion and characteristic classes associated with non-trivial fibrations as in the magnetic monopole.
L. Baulieu, B. Grossman / Constrainedsystems
331
M o s t of this w o r k was c o m p l e t e d while one of us (L.B.) was visiting Rockefeller U n i v e r s i t y in the fall of 1984. H e thanks Professor N. K h u r i for the h o s p i t a l i t y a n d s u p p o r t o f the Rockefeller theoretical physics group. W e w o u l d also lie to t h a n k P r o f e s s o r J a m e s Stasheff for his interest and n u m e r o u s discussions.
Note added in proof A f t e r this w o r k was c o m p l e t e d we b e c a m e aware of two related p a p e r s b y H e n n e a u x [18] a n d M c M u l l a n [19].
Appendix A NILPOTENCY OF S
- : { ,~. /:~ } :c,e. + ¼::a,{ ::~. : j ' }
c°':'a,.
+ f,a~'c'~c#f v,~P'c"'d#, .
(A.1)
F r o m (2) we have that
(cO,o. c% } =zo;::,. { eOa, ( eo,~,coo} } c% "ca=O
(A.2)
by Jacobi identity
(A.3)
= { ~., L~'*, } :::
= { ~,,. L ; } q,pc°c ~
(A.4)
+L;c"c"I. :c % ( COs, f,¢~' ) c%'~c~ -f,#°c"caf,,'t°c '~'= O.
(a.5:
Finally,
c'~c#d.t( f~a, f~'~, } c"'c#'d~,, =
0
(A.6)
* While eq. (A.5) is true for the cases considered in the text, it is not true in the generic case, e.g. supergravity and a relativistic membrane [17]. We thank Dr. M. Henneaux for pointing this out to us.
332
L. Baulieu, B. Grossman / Constrained systems
by symmetry. From (2)
(2)
{ ¢,~,¢/~ } = f¢/i~v,
(:~,0.(:.:)) = (:%. :-,~)+~ +f'Pv{ f'oo, fir },
(a.7)
-((:.:u}:) eo~},eop } byJacobi.
+{{fj,
(a.8)
Multiply by grassmannian c, d:
c~cOdo{fa.°,f<#i.:} d:~'c"cp
:c.c.,°-:o,.( iu.+.) + ((+'.i.:):) + { { fn/', q¢~} q>tl } l c<~ctl
=::a.[-lo,,{ ij.~>'}
(A.9)
+
{ iU::..~> °)
-
{ iUio',, ~,/'}] <°<"
(A.IO)
= cacOd. [f.arf~o'f,", - fn°'fn/'f<,.~,
-fnp'f#"~'f,~.~, + fa/'f,~°~'fl~..i, + f,~"'~f,/'flt..~,] c<'clt =0.
(A.11) Appendix B
BRS I N V A R I A N C E OF H
We define the following quantities: qA=( X q~~ ) , x a = ( X ~ )- x o
~rA= (p:r~),
,
a° =
(B.O
(B.2)
(,~o¢o),
(B.3) 14= 14o + % v ; ¢ -
{ %x a , o : ~ - ~ <1: :r r b
~
c d
n ),
(B.4)
333
L Baulieu, B. Grossman / Constrained systems
with V"b and U~be determined by V~B and f~ar with the appropriate components equal to zero. ~2 = Gb~ b - ~1 U h~d p b)),))~,
(B.5)
{~2, H ) ='O'~{G,,, Ho} + ~(Ho, Ub,.a} Pfll"~l a + G~V"b~) b 1 lib
-
UbcdO~b~CVde'lje
c dr~ I/'a
+ ~ "~ cd~l rl ~ ~-- b
- c2, { ~2, V#b ) .qb.
(8.6)
We use (B.7)
{ G,,, H o ) = - V~bGb,
(8.8)
( ~, H } = ~ U%~oV°?~,7 ~- ~ U , } % V / ¢ , f -- Ubcd°~b]]CVde~ e + Ubceg~@b~C~] d
+ { o,, v~ )e~n'n~- °L{ ~, vo~} ¢
(B.9)
__I -- ~6~a{ UCde , Vab } 6~c~dne,b.
We have the following identity
( H o, {G#, Gb} ) -- ( n o, U , b " ) G , + U,,b"{Ho,G,} =(Gb,(G~,Ho}}+(G,,,(Ho,Gb} = - ( G b, V~CG~) + (O,,, Vb~G,.} =
_
(B.10) )
(B.11) (B.12)
UhJG~Vo~ + U~fV~G~
- { G b, V J ) G ~ +
{G,,, Vb~}O~
= UabCVcdGd q- ( Ho, U,6 c } Go,
(B.13) (8.14)
{ Ho, Cobc } = - U b f V J + U o j V / - uo#v;
- { Gb, V~~ } + { G o, V~c }.
(B.15)
(B.15) serves as an integrability condition resulting from the Jacobi identity, i.e.
334
L. Baulieu, B. Grossman / Constrainedsystems
insist that ( [2, H o } = - ( J2, P . V ~ b ) such that { H o, G. } = Vh.Gh. Then the above identity must hold. By symmetry the following must be zero.
( Vo/, V; } - ( vo;, vj } + (u~/, v;} - { <,:, v/ ) --~ ( Ubj, Vae } -- ( Ubde, V/ } -- ( Ub/, Vde } "q- ( Ubae, V j ) --{UaJ, Vbe}--[-(Udae V/}--(Ud/,Vae}'~-(Ubde,
V/}=O.
(U.16)
Similarly for ( f-~v, f-'a'Y') = 0.
Appendix
C
GAUGE INVARIANCE OF fjo,o Suppose X = X°( p, q) + p 17"~(p, q) 71b ,
(c.1)
H = H°(p,
(c.2)
q) + 6).V"b~b,
V = constant. Supressing indices, we have: {H, X} = 0 =, (a) (b)
( H °, X ° ) = 0 ,
(C.3a)
~2(H°,~}~ + ~[V,~]~ =0
(C.3b) (C.4)
i.e. (b) ~ ( H °, ~" } = - V~',
(G, (u°, P}} = -(G, Vf/} = ( < (/~°,G}) + {H°, (C,~}) = - (G, v ) ~ -
V(G,f~)
(c.5) (c.6)
= (~,v~) + (H°,(6,~)}.
(c.7)
(H°,(G,I/))
(C.8)
Then =0,
335
L. Baulieu, B. Grossman / Constrained svstems
implies ( G, ~" ) = const = 0,
(~, x )
(C.9)
= {6, x ° ) n + a v n + ( v , x °) ~
+ ~(a,
+ WPrl~'r/+ 6)lT:wrlrl,
P)~ (C.10) (C.11a)
( ao, x ° } = - a f ' , (U, X ° ) = V U - V~'+ 17"U+ (G, I7:) - (G, 1,7"),
(6o, x } = (Go, x ° ) = - G f <
(C.11b)
(c.a2)
Eq. (C.11b) is just an integrability condition for (C.11a) and proved by the Jacobi identity. But this is equivalent to X = T ° + T"G,,
(C.13)
with
(G~,X°)=(6~,T°6°)=-6~.
(C.14)
By the above, choose V = V. This choice for V means (3) satisfies an integrability so Z given, V, we can solve for T ". QED. References [1] P.A.M. Dirac, Lectures on quantum mechanics (Yeshiva University, New York 1964) [2] L. Faddeev and V.N. Popov, Phys. Lett. B25 (1967) 30 [3] D. Gross and R. Jackiw, Phys. Rev. D6 (1972) 477; W. Bardeen, Phys. Rev. 184 (1969) 1843; P.H. Frampton and T.W. Kephart, Phys. Rev. Lett. 50 (1983) 1343, 1347; L. Baulieu, Nucl. Phys. B241 (1983) 55; B. Zumino, Y.S. Wu and A. Zee, Nucl. Phys. B239 (1969) 1459; E. Witten, Phys. Lett. l17B (1982) 324; L. Alvarez-Gaum~ and E. Witten, Nucl. Phys. B234 (1983) 269; L. Alvarez-Gaum6 and P. Ginsparg, Nucl. Phys. B243 (1984) 499; M. Atiyah and I. Singer, Proc. Nat. Acad. Sciences, USA 81 (1984) 2599; O. Alvarez, I.M. Singer and B. Zumino, Commun. Math. Phys. 96 (1984) 409 [4] E.S. Fradkin G.A. Vilkovisky, Phys. Lett. 55B (1975) 229; E.S. Fradkin T.E. Fradkina, Phys. Lett. 72B (1978) 343; I.A. Batalin E.S. Fradkin, Phys. Lett. 86B (1979) 263 [5] C. Becchi, A. Rouet R. Stora, Phys. Lett. 52B (1974) 344 [6] C. Itzykson J. Zuber, Quantum field theory (McGraw-Hill, 1980) [7] D. Sullivan, Differential computations in topology, Institut des Hautes Etudes Scientifique [8] R. Bott and L. Tu, Differential forms in algebraic topology (Springer, New York, 1982) [9] R. Bott, Differential topology, foliations and Gelfand-Fuks Cohomology (Springer, Berlin 1978); Relatively, groups and topology II (Les Houches 1983) ed. by B.S. De Witt and R. Stora (North Holland, 1984): Lectures given by L. Baulieu and B. Zumino and R. Stora
336
L. Baulieu, B. Grossman / Constrained systems
[10] P. Simms and N. Woodhouse, Lectures on geometric quantization (Springer, Berlin, 1976) [11] V.N. Gribov, SLAC translation 176 (1977); I. Singer, Commun. Math. Phys. 60 (1963) 7 [12] B. Grossman, Phys. Lett. B152 (1985) 93; R. Jackiw, Phys. Rev. Lett. 54 (1985) 159; Y.S. Wu and A. Zee, Phys. Lett. B152 (1985) 98; D. Boulware, S. Deser and B. Zumino, Phys. Lett. B, in press. [13] A. Aboulsaood, Phys. Lett. 125B (1983) 467; Nucl. Phys. B226 (1983) 309; P. Nelson and A. Manohar, Phys. Rev, Lett. 50 (1983) 943; A.P. Balachandran, et. al., Phys. Rev. Lett. 50 (1983) 1553 [14] B. Grossman, J. Stasheff and T.W. Kephart, Commun. Math. Phys. 96 (1984) 431 [15] L. Faddeev, Phys. Lett. 145B (1984) 81; B. Zumino, ITP-UCSB-preprint NSF-ITP-84-150; J. Mickelson, University of Helsinki preprint, HU-TFT-83-57 Commun. Math. Phys., to be published [16] K. Brown, Cohomology of groups (Springer, New York 1982); J. Stasheff, Bulletin AMS 84 (1978) 513 [17] M. Henneaux, Phys. Rev. Lett. 55 (1985) 769 [18] M. Henneaux, Univ. of Texas preprint, to be published in Phys. Reports [19] D. McMullan, Imperial College preprint 83-84/21 (1984)